The principal block of a ${\mathbb{Z}}_{\mathrm{\ell }}$-spets and Yokonuma type algebras

We formulate conjectures concerning the dimension of the principal block of a ${\mathbb{Z}}_{\mathrm{\ell }}$-spets (as defined in our earlier paper), motivated by analogous statements for finite groups. We show that these conjectures hold in certain situations. For this we introduce and study a Yokonuma type algebra for torus normalisers in $\mathrm{\ell }$-compact groups which may be of independent interest.